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Volume 6, Issue 10, October 1994

Hydrodynamic dispersion broadening of a sedimentation front
View Description Hide DescriptionHydrodynamic dispersion is responsible for the spreading of the sedimentation front even in a noncolloidal monodisperse suspension. Measurements of the broadening of the top front observed during sedimentation have been used in determining the hydrodynamic dispersion coefficient. Hindered settling has an opposed effect and leads to the self‐sharpening of the front. Both effects have to be taken into account simultaneously. This Letter provides a simple, but complete determination of the space and time concentration profile and shows that the final front should consist of a steady‐shape profile propagating at constant velocity. With such a solution, the data of Davis et al. [AIChE J. 34, 123 (1988); J. Fluid Mech. 196, 107 (1988)] give hydrodynamic dispersion coefficient five times larger than their former analysis, in agreement with Lee et al. [Phys. Fluids A 4, 2601 (1992)].

Turbulence reduction in a boundary layer by a local spanwise oscillating surface
View Description Hide DescriptionThis Letter describes an experimental investigation of the response of a turbulent boundary layer on a flat plate to a local spanwise oscillation of the wall, with a nondimensional frequency f ^{+} varying between 0.0033 and 0.0166. The investigation has been carried out for a wall motion amplitude Δz ^{+}=160. The three components of the turbulence intensities and the Reynolds stress prove to be a decreasing function of the frequency. This reduction affects almost the whole boundary layer in a cross section located at the middle of the oscillating wall. The mean streamwise velocity Ū is reduced throughout the region y ^{+}<30. The velocity profiles exhibit a well‐defined log region when plotted against y, nondimensionalized with the friction velocity of the unperturbed boundary layer. The weighted probability density functions of u and v exhibit an increase in intensities of wallward motion related to changes in the structure of the oscillatory flow. The fine structure of the turbulence is also affected by the spanwise oscillation as shown by the reduction of Taylor’s microscale.

A note on the spectra and decay of rotating homogeneous turbulence
View Description Hide DescriptionIn this Letter, the dynamics of spectral energy transfer in rotating homogeneous turbulence is investigated. It is shown that the wave number k _{Ω}=(Ω^{3}/ε)^{1/2} (defined by the rotation speed Ω and dissipation ε) determines the turbulence length scale, above which rotation effects on spectral transfer and on energy spectrum form are important. From the rotation‐modified spectrum, turbulence decay laws are also inferred.

The spreading of drops with intermolecular forces
View Description Hide DescriptionThe spreading of a thin drop of fluid that partially wets a plane surface is calculated. Capillarity, slip, and intermolecular forces are included in the model, and it is shown that a complete solution is possible without having to exclude the vicinity of the contact line and without having to assume the dynamic behavior of the contact angle. An equation is found for the evolution of the drop radius; when the drop is not close to its equilibrium radius, the spreading law has the expected one‐tenth power dependence on the radius, with a coefficient which is determined as a function of the intermolecular forces and the slip coefficient. The calculation is performed for small static contact angles and also for the limit when this angle is zero.

Fractal properties of flocs formed by fluid shear and differential settling
View Description Hide DescriptionThe fractal properties of fine‐grained sediment flocs formed by fluid shear and by differential settling were studied through floc porosity–size relationships. Flocs were produced from flocculation tests in a Couette‐type flocculator in which fluid shear was the dominant collision mechanism for flocculation, and in a disk‐type flocculator in which differential settling was the dominant collision mechanism. Floc samples were then introduced into a settling tube to measure their sizes and settling speeds by a double‐exposure photographic method. The porosities of flocs were determined from the settling speed data using a porous sphere settling model. It was found that the flocs produced in the Couette‐type flocculator could be regarded as fractals, with a fractal dimension ranging from 1.83 to 1.97. However, the flocs produced in the disk‐type flocculator did not exhibit simple scaling behavior and could not be regarded as fractals. The differences in fractal property are due to the differences in floc formation schemes associated with the collision mechanisms. The similarities between experimentally obtained flocs and computer simulated clusters are discussed. The multistage growth model, a formation scheme for flocs formed by fluid shear, is consistent with the cluster–cluster model used in computer simulations. A formation scheme for flocs formed by differential settling is postulated; it is consistent with the particle–cluster model used in computer simulations.

Stability of a layer of viscous magnetic fluid flow down an inclined plane
View Description Hide DescriptionThis paper concerns the linear stability of a layer of viscousmagnetic fluidflow down an inclined plane under the influence of gravity and a tangential magnetic field. The stability of a magnetic fluid in a three‐dimensional space is first reduced to the stability of the flow in a two‐dimensional space by using Squire’s transformation. The stability of long waves and short waves is analyzed asymptotically. The stability of waves with intermediate length is obtained numerically. It is found that the magnetic field has a stabilizing effect on both the surface and shear modes and can be used to postpone the instability of such flows.

Curvature effects on axisymmetric instability of conduction regime in a tall air‐filled annulus
View Description Hide DescriptionThis paper numerically studies curvature effects on the instability of the conduction regime of natural convection in a tall air‐filled differentially heated annulus of vertical aspect ratio 16 by integrating the two‐dimensional axisymmetric Navier–Stokes equations in the Boussinesq approximation. The numerical algorithm combines a pseudospectral Chebyshev space discretization with a second‐order time‐stepping scheme. It is shown that, in contrast with linear stability analysis of the conduction solution, the time‐periodic cross‐roll instability does not take place in finite aspect ratio cavities for small values of the radius ratio. For all values of the radius ratio transition to unsteadiness occurs through supercritical Hopf bifurcations. Extensive computations show very complex behaviors of the unsteady solutions depending on the radius ratio. The nature of the reverse transition to steady state that occurs for increasing value of the Rayleigh number is also found to depend strongly on the value of the radius ratio.

Nonlinear evolution equations for thin liquid films with insoluble surfactants
View Description Hide DescriptionThe dynamics of a free‐liquid film with insoluble surfactants is followed until film rupture with a simple model based on three nonlinear evolution equations for the film thickness, the surfactants concentration and the tangential velocity of the fluid in the film. This model is derived asymptotically from the full Navier–Stokes equations for free films and incorporates the effect of van der Waals attraction, capillary forces and Marangoni forces due to gradients of surface tension. Different stability regimes are observed numerically for periodic and fixed boundary conditions and several initial conditions. Furthermore, the role of the relevant parameters (Hamaker constant, tension, Marangoni number) on the rupture time is assessed and comparison is made with the flow dynamics for a liquid film with insoluble surfactants on a solid substrate.

The influence of surfactant on the bubble motion in Hele–Shaw cells
View Description Hide DescriptionThe surfactant influence on the bubble motion in a Hele–Shaw cell was studied experimentally. In order to differentiate the cases with and without the surfactant influence, the motion of air bubbles and waterdrops driven by silicone oil was investigated. The waterdrops contained a predetermined amount of an organic surfactant(sodium dodecyl sulfate), so that the dependence of their motion on the surfactant concentration could be studied systematically. In case of air bubbles in silicone oil, surfactants were likely to have negligible influence, although they might be present as contaminants, and their translational velocities were observed to be close to the prediction of Taylor and Saffman [Q. J. Mech. Appl. Math. 12, 265 (1959)]. The bubble shapes were also in accordance with available theories for a surfactant‐free system. The waterdrops, on the other hand, behaved very differently, in that the translational velocities were smaller by an order of magnitude and their shapes were very unusual. These observations are apparently consistent with those of Kopf‐Sill and Homsy [Phys. Fluids 31, 18 (1988)], and the present study suggests that the perplexing observations by them may be due to the influence of surface‐active contaminants.

Concentration waves and flow modification in a particle‐laden circular vortex
View Description Hide DescriptionEvolution of a two‐dimensional axisymmetrical vortex laden with solid heavy particles is studied analytically and numerically. The particulate phase is assumed to be dilute enough to neglect the effects of particle–particle collisions. Only sufficiently small particle Stokes (St) and Reynolds numbers are considered, for which an approximate solution for the particle velocity can be derived. An analytical solution to a Cauchy problem is obtained for initially uniform concentration of particles in a circular flow describing the accumulation of particles in the form of a kinematic wave and the corresponding modification of the carrier flow. According to this solution, a steep peak of the concentration develops forming the wave crest which propagates out of the vortex. Due to the interaction between the two phases, a fluid velocity component directed towards the vortex center is generated, so that in the vicinity of the crest the vortex acquires a spiral‐like shape. At later stages, the growth of the crest is inhibited and its propagation velocity decreases. Analysis of the problem for particles with larger Stokes numbers shows that the accumulation process is most intense when St is close to a critical value St_{*} which generally depends on the vortex structure and, for the flow considered, is of the order unity.

Flow regimes in two‐dimensional mixed convection with spatially periodic lower wall heating
View Description Hide DescriptionTwo‐dimensional numerical simulations of flow through a channel with spatially periodic temperature boundary conditions at the lower wall have been carried out. A spectral method with a Fourier series expansion in the streamwise direction and a Chebyshev expansion in the vertical direction is employed. A bifurcation to a limit cycle occurs at Reynolds numbers as low as Re=4 and Rayleigh numbers as low as Ra=14 500. The frequency of oscillation decreases with Re and is essentially independent of Ra. Maps of flow regimes for varying periodicities have been created. Regimes with periodicities longer than the imposed temperature boundary condition have been found.

Oscillatory buoyant thermocapillary flow
View Description Hide DescriptionA computational study of the character and stability of two‐dimensional buoyant thermocapillary flows, valid to leading order in capillary number (Ca), is conducted in the Grashof number (Gr), Reynolds number (Re), aspect ratio, and Prandtl number (Pr) parameter space. Calculations of thermocapillary convection for low Pr fluids have generally produced steady results. Calculations of pure buoyant convection (Re=0) exhibit a Hopf bifurcation at Gr_{ cr } (no thermocapillarity) that is well understood. Thus, the combined thermocapillary buoyant problem is studied to investigate the onset of oscillatory convection in the limit Gr→0. The unsteady natural convection pattern at fixed Gr≳Gr_{ cr } is modified only slightly for low values of Re. When thermocapillarity acts in conjunction with buoyancy (Re≳0) it is stabilizing, in that the transition to unsteady flow occurs at Gr≳Gr_{ cr }, as defined for the strictly buoyant problem. When thermocapillarity acts in opposition to buoyancy (Re<0), it is destabilizing for relatively small values of ‖Re‖, but thermocapillarity ultimately dominates the convective pattern for larger ‖Re‖, and the resulting flow is steady for the range of parameter values considered. Stability boundaries for the onset of oscillatory convection in the Gr–Re plane are given for representative values of the cavity aspect ratio and Pr.

Atmospheric interfacial waves in the presence of two moving fluid layers
View Description Hide DescriptionAtmospheric waves at the interface between two flowing layers of air are studied in this paper. The lower layer is assumed to be incompressible and to flow irrotationally, and its motion might be the result of a distant thunderstorm, for example. The upper layer is modeled as a compressible isothermal atmosphere, so that if it were stationary, its density and pressure would both decrease exponentially with height. The equations of motion in the upper layer are linearized under the assumption that the lower layer of incompressible fluid is ‘‘thin’’ (its weight is a small fraction of the total), but the possibility of large‐amplitude disturbances at the interface is nevertheless allowed. A linearized theory of wave propagation in this system is discussed, and a numerical scheme is outlined for the solution of the nonlinear equations. The results confirm the predictions of a model of Forbes and Belward [Phys. Fluids A 4, 2222 (1992)], in which the upper atmosphere was assumed stationary, and demonstrate that this simpler model gives results that are likely to be useful over most of the range of values of the speed in the upper layer encountered in practice. Nonlinear waves near the limiting height are discussed, and a very significant qualitative difference between the predictions of the linearized theory and the nonlinear results concerning progressive waves is analyzed, and may be of importance in meteorology.

The modulation of short waves riding on solitary waves
View Description Hide DescriptionThe modulation of linear short waves riding on a long finite‐amplitude solitary wave has been analyzed numerically. It is found that the maximum modulated wave number, frequency, and amplitude of short waves always occur at the crest of solitary waves. This paper shows that the modulated wave number on the crest of solitary waves increases significantly as the amplitude of the solitary waves increases, and that the modulated short wave frequency and amplitude on the crest increase almost linearly.

Rayleigh–Taylor and shear driven mixing with an unstable thermal stratification
View Description Hide DescriptionA new water channel experiment has been used to study turbulent mixing driven by buoyancy, and by combined buoyancy and shear. Density differences were produced by thermal stratification. The experiment was statistically steady, and a space–time transformation in the streamwise direction permitted a continuous study of the mixing evolution. Dye and digitized photographs were used to study the mixing process. An ensemble average of images gave the average mixing layer growth rate and the distribution of light and heavy fluid in the mixing layer. The structure of the early growth of buoyancy dominated mixing and of combined shear and buoyancy mixing is presented. The mixing transition from combined shear and buoyancy mixing to buoyancy dominated mixing occurred at Richardson numbers from −5 to −11. It was found that buoyancy dominated a self‐similar mixing stage for the range of flows (ΔU=0 to 2 cm/s) and density differences (Δρ=0.38 to 2.4 kg/m^{3}). Transition to self‐similar mixing occurred at a Reynolds number from 670 to 1200. The self‐similar mixing width for all tests had a quadratic growth rate with an average acceleration constant of 0.070 and a standard deviation of 0.011.

The existence of critical Reynolds numbers in pipe entrance flows subjected to infinitesimal axisymmetric disturbances
View Description Hide DescriptionA modified Chebyshev operational Tau approach was implemented to establish temporal eigenvalues for the axisymmetric Sexl stability equation, transformed to remove the singularity at the origin. Its application to the pipe entrance flow stability problem resulted in the determination of a well‐defined maximum axial position—relatively close to the pipe inlet—which possesses a critical Reynolds number of infinity. Thus for the first time a whole range of pipe entrance velocity profile shapes is established—of which the Hagen–Poiseuille limit is only one—for which critical Reynolds numbers do not exist. When combined with the fact that nonparallel effects disappear at the last linearly unstable station, this result is significant because it shows that the greater proportion of the pipe entrance requires finite amplitude disturbances to destabilize the flow.

Propagation and transport properties of dipolar vortices on a γ plane
View Description Hide DescriptionThe dynamics and transport properties of dipolar vortices on a γ plane (a plane where the Coriolis parameter has a quadratic variation with the latitude) are studied using the modulated point‐vortex model. Similarly to the β‐plane case, different regimes are found for the evolution of a single dipole, depending on the initial direction of propagation α_{0}. Two steadily translating couples exist: The one rotating eastward (α_{0}=0) has a stable trajectory and the one rotating westward (α_{0}=π) is unstable. For initial angles in the range 0<α_{0}<π, the couple moves along sine‐like, 8‐shaped and cycloid‐like trajectories. In all solutions the dynamically relevant variables (the latitude and the direction of propagation) are periodic. The advection equations of passive particles in the dipole’s velocity field can be exactly written in the form of a periodically perturbed integrable Hamiltonian system. The study of transport is performed using a ‘‘dynamical‐systems theory’’ approach. The entrainment and detrainment of fluid as a function of γ and α_{0} are computed exactly from some invariant curves in the Poincaré map and approximately by using the Melnikov function. The exchange of mass increases with both increasing γ and α_{0}, while the rate at which this occurs has a maximum for some α_{0} and increases with γ. A major difference in particle spreading exists between dipoles which exactly return to their initial position after an integer number of oscillations and dipoles that do not. In the former case the Poincaré map shows broad areas of unstirred fluid coinciding with the maximum radial displacement in the dipole’s meandering trajectory.

Effects of curvature variations on the nonlinear evolution of Goertler vortices
View Description Hide DescriptionThe nonlinear development of Goertler vortices over walls of variable curvature is studied. The parabolized disturbance equations governing the problem for small curvature, high Reynolds number, and order unity Goertler number are integrated numerically. Cases with concave, convex, and zero curvatures are analyzed. The ‘‘mushroom‐shaped’’ distributions of low‐momentum fluid riding above high‐momentum fluid that are subject to secondary instability are predicted. The results show significant stabilization of disturbances introduced from a concave into a convex region, where new sets of vortices are successively created with opposite rotation to the preceding set. The convex curvature tends to eliminate the inflection points from the spanwise and normal profiles of the streamwise velocity and, hence, suppresses the oscillatory secondary instability that leads to turbulence. Decay of Goertler vortices in a flat region is found to be less rapid than in a convex region.

Boundary layer receptivity to free‐stream vorticity
View Description Hide DescriptionThe receptivity to free‐stream vorticity of the boundary layer over a flat plate with an elliptic leading edge is investigated numerically by solving the incompressible Navier–Stokes system in general curvilinear coordinates with the vorticity and streamfunction as dependent variables. A small‐amplitude vortical disturbance is introduced at the upstream boundary and the governing equations solved time accurately to evaluate the spatial and temporal growth of the perturbations leading to instability waves [Tollmien–Schlichting (TS) waves] in the boundary layer. The effect of disturbance amplitude, orientation, and the effect of the leading edge and of surface curvature are investigated for the case of spanwise vorticity. Simulations reveal, for the conditions considered, a linear variation in the TS response with forcing amplitude for perturbations of free‐stream velocity that are either symmetrical or asymmetrical with respect to the basic‐state stagnation streamline. The presence near the leading edge of a large, oscillating component of velocity normal to the airfoil axis for the case of asymmetrical forcing results, for the same strength of input disturbance, in an increase in the TS response aft of the juncture and in the appearance of a superharmonic component of the disturbance motion near the tip of the nose. This superharmonic decays rapidly in the streamwise direction. In all cases considered, the first clear appearance of the TS mode occurs aft of the surface pressure‐gradient maximum. Changes to the geometry that increase the maximum in steady surface pressure gradient are found to increase receptivity.

Turbulence characteristics in cylindrical liquid jets
View Description Hide DescriptionA study has been made of the flow patterns and turbulence characteristics in free liquid jets in order to determine the rate of decay of turbulence properties along the jet. Mean streamwise velocities and streamwise velocities and streamwise and cross‐streamwise turbulence intensities were measured using laser Doppler velocimetry. The jet Reynolds number was varied between 1000 and 30 000, with the diameter of the liquid jet D=3.051 mm. Using a power law model for the time decay of turbulence kinetic energy, it was found that turbulence decays, on average with an exponent N=1, independent of the Reynolds number. A constant power for the decay implies Reynolds number similarity throughout this range. Substantial reductions in the degree of anisotropy occur downstream from the injector exit as the jet relaxes from a fully developed turbulent pipe flow profile to a flat profile. For the intermediate range of Reynolds numbers (10 000–20 000), the relaxation distance was 20D, almost independent of the Reynolds number. At high values of Reynolds number (20 000–30 000), the relaxation process was very fast, generally within three diameters from the injector exit.